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High-order harmonic generation driven by metal nanotip

photoemission: theory and simulations

M. F. Ciappina1, J. A. Perez-Hernandez2, M.

Lewenstein3,4, M. Kruger5, and P. Hommelhoff5

1Department of Physics, Auburn University, Auburn, Alabama 36849, USA

2Centro de Laseres Pulsados (CLPU),

Parque Cientıfico, 37185 Villamayor, Salamanca, Spain

3ICFO-Institut de Ciencies Fotoniques,

Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain

4ICREA-Institucio Catalana de Recerca i Estudis Avancats,

Lluis Companys 23, 08010 Barcelona, Spain and

5 Department of Physics, Friedrich-Alexander-Universitat Erlangen-Nurnberg,

Staudtstr. 1, D-91058 Erlangen, Germany,

and Ultrafast Quantum Optics Group,

Max-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Str. 1,

D-85748 Garching bei Munchen, Germany

1

Abstract

We present theoretical predictions of high-order harmonic generation (HHG) resulting from the

interaction of short femtosecond laser pulses with metal nanotips. It has been demonstrated that

high energy electrons can be generated using nanotips as sources; furthermore the recollision mecha-

nism has been proven to be the physical mechanism behind this photoemission. If recollision exists,

it should be possible to convert the laser-gained energy by the electron in the continuum in a high

energy photon. Consequently the emission of harmonic radiation appears to be viable, although

it has not been experimentally demonstrated hitherto. We employ a quantum mechanical time

dependent approach to model the electron dipole moment including both the laser experimental

conditions and the bulk matter properties. The use of metal tips shall pave a new way of generating

coherent XUV light with a femtosecond laser field.

PACS numbers: 42.65.Ky,78.67.Bf, 32.80.Rm

Keywords: high-order harmonics generation; nanotips; plasmonics

2

One of the most prominent examples of the nonlinear interaction between laser and

matter, atoms and molecules, is high-order harmonic generation (HHG) process [1, 2]. The

great interest of this phenomenon resides in the fact that HHG represents one of the most

reliable pathways to generate coherent ultraviolet (UV) to extreme ultraviolet (XUV) light.

In addition, HHG based on atoms and molecules has proven to be a robust source for

the generation of attosecond pulses trains [3], that can be temporally confined to a single

XUV attosecond pulse, now with high repetition rates [4–7]. Thanks to its remarkable

properties, HHG can as well be employed to extract temporal and spatial information with

both attosecond and sub-Angstrom resolution on the generating system [8]. Furthermore,

it represents a considerable tool to scrutinize the atomic world within its natural temporal

and spatial scales [9–14].

Instead of using atoms and molecules in the gas phase as active medium, the utilization of

bulk matter, for instance metal nanotips [15], nanoparticles [16] or ablation plumes [17, 18],

has recently been put forward (see e.g. [19] for more references). For instance, the laser

matter phenomenon called above-threshold photoemission (ATP) represents the counterpart

of the above-threshold ionization (ATI) in atoms and molecules, but the underlying physics

is much richer and quite different in nature (see e.g. [20, 21] for field-enhancement and

near-field effects). Several models have been recently developed and applied in order to

understand the experimental data and to guide future measurements [22–25]. Furthermore,

it has been demonstrated that solid state samples can also be used as a generators of high

order harmonic radiation, although the HHG phenomenon using bulk matter is at its very

beginning, both theoretically and experimentally [26, 27].

Another related process, combining noble gases and bulk matter, is the generation of

harmonic radiation using (plasmonically) enhanced fields. The first demonstration of such

an effect was obtained employing surface plasmonic resonances that can locally amplify

the incoming laser field [28]. Using such resonances, amplifications in intensity greater

than 20 dB can be achieved [29, 30]. Consequently, when a low intensity femtosecond

laser pulse couples to the plasmonic mode of a metal nanoparticle, it initiates a collective

oscillation among free electrons within the metal. A region of highly amplified electric field,

exceeding the threshold for HHG, can so be generated while these free charges redistribute

the response field around the metal nanostructure. By injection of noble gases surrounding

the nanoparticle high order harmonics can be produced. Particularly, using gold bow-tie

3

shaped nanostructures it has been demonstrated that the initially modest laser field can

be amplified sufficiently to generate high energy photons, in the XUV regime, and the

radiation generated from the enhanced laser field, localized at each nanostructure, acts as

a point-like source, enabling collimation of this coherent radiation by means of constructive

interference [28]. Recently there has been extensive theoretical work looking at HHG driven

by spatially nonhomogeneous fields [31–40]. However, the initial sudden excitement about

the utilization of plasmonic fields for HHG in the XUV range was put in debate by recent

findings [41–43]. Fortunately, alternative ways to enhance coherent light were explored

(e.g. the production of high energy photoelectrons using enhanced near-fields from dielectric

nanoparticles [16], metal nanoparticles [44–46] and metal nanotips [15, 19–21, 47–53].

In this contribution we predict that it is entirely possible to generate high-order harmonics

directly from metal nanotips. We employ available laser source parameters and model the

metal tip with a fully quantum mechanical model. As it is well known, the main physical

mechanism behind the generation of high order harmonics is the electron recollision and

consequently the model used should include it. It was already shown that the recollision

mechanism is also needed to describe above-threshold photoemission (ATP) measurements

and, considering these two laser-matter phenomena, i.e. the photoemitted electrons and the

high frequency radiation, are physically linked, we could argue that metal nanotips can be

used as sources of XUV radiation as well.

The theoretical model we use here has already been described elsewhere and employed for

the calculation of the electron photoemission in metal nanotips [19, 48]. As a consequence we

only give a brief overview and we emphasize the numerical tools needed to compute the HHG

spectra. In short, the one dimensional time dependent Schrodinger equation (1D-TDSE) is

solved for a single active electron in a model potential. We employ a narrow, few atomic

units wide, potential well with variable depth (W +EF , where W is the work function and

EF the Fermi energy) to model the metal surface. This depth and width of the well are

chosen in such a way as to match the actual metal tip parameters. In our case we employ

the parameters for clean gold, i.e. W = 5.5 eV and a EF = 4.5 eV, but other typical metals,

as tungsten, can be used as well. The ground state of the active electron represents the

initial state in the metal nanotip. The electronic wavefunction is confined by an infinitely

high potential wall on one side and on the other side by a potential step representing the

metal-vacuum surface barrier. In addition we consider an image-force potential that gives

4

a smoother shape to the surface barrier potential. The evanescent part of the electronic

wavefunction penetrates into the classically forbidden (vacuum) region. The rescattering

mechanism, mainly responsible of the high energy region of the photoelectron spectra and

the high order harmonic generation, is closely linked with this evanescent behavior.

We employ laser pulses of the form EL(t) = E0 f(t) sin(ωt+φCEP ), with E0, ω and φCEP

are the laser electric field peak amplitude, the laser frequency and the carrier envelope phase

(CEP), respectively. The envelope f(t) can be chosen between sine-squared or trapezoidal

shape. The former allows us to model ultrashort laser pulses, 2-4 cycles long, and to study

the effects of the φCEP in the HHG spectra. On the other hand, the latter is needed in

order to model long pulses. In our computations we employ a pulse 10 full cycles long with

2 cycles of turn on and off and 6 cycles of constant amplitude. In addition to the laser

electric field, we include a static field Edc that arises due to the tip bias voltage. As a

consequence we can write the potential the electron feels, without considering the image

force, as V (z, t) = (Edc + EL(t))z. The electronic wavefunction is time propagated using

the Crank-Nicolson scheme under the influence of the external fields. Finally, the harmonic

spectra are retrieved by Fourier-transforming the dipole acceleration that is obtained from

the electronic dipole moment (for details see e.g. [54, 55]).

We first study the CEP dependence of the harmonic spectra generated when a metal

(Au) nanotip is illuminated by a femtosecond laser pulse. The experimental confirmation of

our predictions could be done by employing a similar experimental set-up as the one used

for electron photoemission, although some questions remain to be solved [15, 19, 47]. For

instance, we are unable in our calculations to estimate absolute values for the harmonic

signal. In addition, we model a single atom response. Propagation and mode-matching

effects are completely neglected in our approach. These last could play a role, although

it has been argued in two recent experiments [28, 56] that in the harmonic emission from

nanosources one could safely ignore them due to the strong confinement of the radiation

sources in dimensions smaller than the laser wavelength.

We use ultrashort laser pulses with a full width at half maximum (FWHM) duration of

2 fs and 4 fs (corresponding approximately to a 2 and 4 total cycles), wavelength λ = 800

nm (photon energy 1.55 eV) and we vary the carrier envelope phase φCEP in order to cover

its full range ([0, 2π]). We model laser peak fields of up to 20 GV m−1, as this is close to the

experimentally accessible range before damage may set in. Furthermore, plasma effects do

5

not appear whenever the laser intensity does not exceed the saturation limit of each specific

chemical specimen target (gold, in this case) and for the mentioned values we are below such

limit. Note that the values of the peak electric fields mentioned here mean the physical fields

at the tip, including the effects of field enhancement. For the static field Edc we employ a

typical value, Edc = −0.4 GV m−1 and also Edc = +2 GV/m, corresponding to a positive

tip bias voltage.

In Fig. 1 we show calculations of the harmonic yield as a function of the harmonic order

and φCEP for the 2 fs FWHM laser pulse case; Fig. 2 presents the 4 fs FWHM laser pulse

case. The different panels correspond to a peak electric field E0 = 10 GV m−1 and a static

field Edc = −0.4 GV m−1 (Fig. 1(a)); E0 = 10 GV m−1 and Edc = +2 GV m−1 (Fig. 2(a));

E0 = 20 GV m−1 and Edc = −0.4 GV m−1 (Fig. 2(a)) and E0 = 20 GV m−1 and Edc = +2

GV m−1 (Fig. 2(b)). The first feature we can observe is the strong modulation in the spectra

as the φCEP changes, more pronounced for the 2 fs FWHM case (similar behavior is observed

in atoms, see e.g. [7, 57]). A clear harmonic cutoff can be seen for the 2 fs FWHM case

at nc ≈ 5 (equivalent to a photon energy of 7.75 eV) for E0 = 10 GV m−1 (Fig. 1(a) and

Fig. 1(b)) and one at nc ≈ 10 (equivalent to a photon energy of 15.5 eV) for E0 = 20 GV

m−1 (Fig. 2(a) and Fig. 2(b)), although the latter is more visible for particular values of the

φCEP . When we increase the pulse length to 4 fs FWHM the modulation is less noticeable,

but nevertheless harmonic cutoff values similar to the previous cases are observed. For

Edc > 0 an increase in the harmonic yield is observed for some values of φCEP and it is more

clear for the cases when E0 = 20 GV m−1. This feature could be experimentally exploited

as a larger signal will be easier to detect.

We can consider the semiclassical simple man’s model predictions [58, 59] in order to

characterize the harmonic cutoff, although the influence of the static field Edc will not be

included. It is well established that for atoms and molecules nc = (3.17Up+ Ip)/ω, where nc

is the harmonic order at the cutoff and Up the ponderomotive energy (Up = E2

0/4ω2) [54] and

Ip the ionization potential of the atomic or molecular species under consideration. Inserting

the values of the peak electric field and laser wavelength and using an equivalent Ip equal to

the metal work function W we can corroborate the cutoff values obtained from our quantum

mechanical model. For instance, for E0 = 10 GV m−1 (0.02 a.u) nc ≈ 5 and this value

is in very good agreement with the quantum mechanical calculations (see. e.g. Fig. 1(a),

Fig. 1(b), Fig. 2(a) and Fig. 2(b)). For positive values of Edc it appears that the harmonic

6

cutoff increases, but the effect could be masked by the CEP influence. For longer pulses,

however, we do observe a clear harmonic extension for Edc > 0 (see below).

Next, in Fig. 3, we show harmonic spectra by using a long (10 cycles) trapezoidally

shaped laser pulse of λ = 685 nm (photon energy 1.81 eV). The different panels correspond

to various values of the peak electric field E0, namely 10 GV m−1, 15 GV m−1 and 20 GV

m−1 for Fig. 3(a), Fig. 3(b) and Fig. 3(c), respectively. In the three panels the magenta/dark

gray (green/light gray) represent the case of Edc = −0.4 GV m−1 (Edc = +2 GV m−1). We

can observe an increasing of the relative yield in the plateau region for positive values of

the Edc field. This gain in conversion efficiency is again important for ease of experimental

radiation detection.

Finally we compute in Fig. 4 harmonic spectra by using a long (10 cycles) trapezoidal

shaped laser pulse of λ = 1800 nm (photon energy 0.69 eV) and a peak electric field E0 = 10

GV m−1. The magenta/dark gray curve is for Edc = −0.4 GV m−1 while the green/light

gray one is for Edc = +2 GV m−1. We can observe harmonic cutoffs of around nc = 30

(that correspond to an equivalent energy of 20 eV) for the former case and nc = 60 (41 eV)

for the latter. In here, the semiclassical model predicts nc ≈ 27 and we could observe that

the influence of the static electric field appears to be more pronounced when λ increases.

Considering the semiclassical cutoff is proportional to λ2 it is evident that the utilization of

longer wavelengths laser sources would allow us to reach high energy photons.

In conclusion, we predict that it is possible to generate high order harmonics directly from

metal nanotips. We employ a quantum mechanical model in order to compute the HHG

yield and we use typical laser parameters available experimentally. A comparable approach

was successfully applied to predict the photoelectron spectra under similar experimental

conditions [19]. We observe a strong modulation of the HHG spectra with the variation of the

φCEP for short pulses and a noticeable extension of the HHG cutoff when negative static fields

are employed, more pronounced when longer wavelengths laser sources are employed. For the

parameters we have used we can safely neglect any spatial variation of the (plasmonically)

enhanced field. For instance, using λ = 1800 nm and E0 = 10 GV m−1 (0.02 a.u.) the

classical electron quiver radius α (α = E0/ω2) is around 1.6 nm and, for typical metal

nanotips with radii between R = 5 nm and R = 50 nm, the estimated optical field decay

length (1/e) is around L = (0.82 ± 0.04)R [60]. Consequently the laser-ionized electron

would practically feel a spatially homogeneous electric field as α ≪ L for the parameters

7

given. The aspect of the spatially inhomogenuous field will become important for other

experimental conditions, which will be considered in a future work. In addition, we plan

to extend the simple man’s model (SMM), already employed for electron emission in metal

nanotips, to treat HHG. In this way, a deeper understanding of the underlying physics of

harmonic emission from metal nanotips will be achieved.

We acknowledge the financial support of the MICINN projects (FIS2008-00784 TO-

QATA, FIS2008-06368-C02-01, and FIS2010-12834), ERC Advanced Grant QUAGATUA,

the Alexander von Humboldt Foundation, the Hamburg Theory Prize (M.L.), and the DFG

Cluster of Excellence Munich Center for Advanced Photonics. This research has been par-

tially supported by Fundacio Privada Cellex. J.A.P.-H. acknowledges support from the

Spanish MINECO through the Consolider Program SAUUL (CSD2007-00013) and research

project FIS2009- 09522, from Junta de Castilla y Leon through the Program for Groups

of Excellence (GR27), and from the ERC Seventh Framework Programme (LASERLAB-

EUROPE, Grant No. 228334). This work was made possible in part by a grant of high

performance computing resources and technical support from the Alabama Supercomputer

Authority.

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FIG. 1. (color online) Contour plots of the high-order harmonic spectra as a function harmonic

order and carrier envelope phase (CEP) for a metal (Au) nanotip using a 2 fs FWHM laser pulse

with a wavelength λ = 800 nm. Panel a) E0 = 10 GV m−1, Edc = −0.4 GV m−1; panel b) E0 = 10

GV m−1, Edc = +2 GV m−1; panel c) E0 = 20 GV m−1, Edc = −0.4 GV m−1 and panel d)

E0 = 20 GV m−1, Edc = +2 GV m−1.

12

FIG. 2. (color online) Contour plots of the high-order harmonic spectra as a function harmonic

order and carrier envelope phase (CEP) for a metal (Au) nanotip using a 4 fs FWHM laser pulse

with a wavelength λ = 800 nm. Panel a) E0 = 10 GV m−1, Edc = −0.4 GV m−1; panel b) E0 = 10

GV m−1, Edc = +2 GV m−1; panel c) E0 = 20 GV m−1, Edc = −0.4 GV m−1 and panel d)

E0 = 20 GV m−1, Edc = +2 GV m−1.

13

FIG. 3. (color online) HHG spectra as a function of harmonic order for a metal (Au) nanotip using

a trapezoidal shaped laser pulse with 10 cycles of total time and a wavelength λ = 685 nm. Panel

a) E0 = 10 GV m−1; panel b) E0 = 15 GV m−1 and panel c) E0 = 20 GV m−1. In all the panels

magenta: Edc = −0.4 GV m−1, green: Edc = +2 GV m−1.

14

FIG. 4. (color online) HHG spectra as a function of harmonic order for a metal (Au) nanotip using

a trapezoidal shaped laser pulse with 10 cycles of total time and a wavelength λ = 1800 nm and

a peak laser electric field E0 = 10 GV m−1. Magenta Edc = −0.4 GV m−1, green Edc = +2 GV

m−1.

15